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Beauty And The Beast Part 2: Apprehending The Missing Supercurrent

Gregory W. Moore, Ranveer Kumar Singh

TL;DR

The paper provides an explicit construction of a dimension-$\tfrac{3}{2}$ supercurrent inside a spin lift of the Moonshine module (Beauty And The Beast) by leveraging Arf theory and error-correcting codes. It develops the necessary VOA/lattice framework, including untwisted/twisted lattice VOAs and their orbifolds, and then implements spin lifts via gauging a non-anomalous $\mathbb{Z}_2$ symmetry to realize a consistent $\mathcal{N}=1$ SCVOA on a subspace of the BB module. By embedding superconformal sublattices connected to binary and quaternary codes, the authors construct multiple inequivalent supercurrents, notably identifying Duncan’s spinor and a Golay-code-based current. The work confirms a rich, yet constrained, supersymmetric structure within the BB module and clarifies its relation to the FLM Moonshine module and the Monster CFT, highlighting profound connections between moonshine, lattice theory, and topological couplings. These results illuminate how code-theoretic data governs superconformal structures in Moonshine-related VOAs and their spin lifts, with potential implications for Moonshine phenomena and higher-supersymmetry searches in related theories.

Abstract

The Moonshine module is a $c=24$ conformal field theory (CFT) whose automorphism group is the Monster group. It was argued by Dixon, Ginsparg, and Harvey in \cite{Dixon:1988qd} that there exists a spin lift of the Moonshine CFT with superconformal symmetry. Reference \cite{Dixon:1988qd} did not provide an explicit construction of a superconformal current. The present paper fills that gap. In fact, we will construct several superconformal currents in a spin lift of the Moonshine CFT using techniques developed in \cite{Harvey:2020jvu}. In particular, our construction relies on error correcting codes.

Beauty And The Beast Part 2: Apprehending The Missing Supercurrent

TL;DR

The paper provides an explicit construction of a dimension- supercurrent inside a spin lift of the Moonshine module (Beauty And The Beast) by leveraging Arf theory and error-correcting codes. It develops the necessary VOA/lattice framework, including untwisted/twisted lattice VOAs and their orbifolds, and then implements spin lifts via gauging a non-anomalous symmetry to realize a consistent SCVOA on a subspace of the BB module. By embedding superconformal sublattices connected to binary and quaternary codes, the authors construct multiple inequivalent supercurrents, notably identifying Duncan’s spinor and a Golay-code-based current. The work confirms a rich, yet constrained, supersymmetric structure within the BB module and clarifies its relation to the FLM Moonshine module and the Monster CFT, highlighting profound connections between moonshine, lattice theory, and topological couplings. These results illuminate how code-theoretic data governs superconformal structures in Moonshine-related VOAs and their spin lifts, with potential implications for Moonshine phenomena and higher-supersymmetry searches in related theories.

Abstract

The Moonshine module is a conformal field theory (CFT) whose automorphism group is the Monster group. It was argued by Dixon, Ginsparg, and Harvey in \cite{Dixon:1988qd} that there exists a spin lift of the Moonshine CFT with superconformal symmetry. Reference \cite{Dixon:1988qd} did not provide an explicit construction of a superconformal current. The present paper fills that gap. In fact, we will construct several superconformal currents in a spin lift of the Moonshine CFT using techniques developed in \cite{Harvey:2020jvu}. In particular, our construction relies on error correcting codes.
Paper Structure (27 sections, 41 theorems, 418 equations, 1 table)

This paper contains 27 sections, 41 theorems, 418 equations, 1 table.

Table of Contents

  1. Introduction
  2. Vertex Operator Algebras And Their Extensions
  3. Definition Of A Vertex Operator Algebra
  4. Representation Of A Vertex Operator Algebra
  5. Extension Of A Vertex Operator Algebra By A Real Hermitian Representation
  6. Lattice Vertex Operator Algebras And Their Orbifolds
  7. Untwisted Vertex Operator Algebra
  8. Twisted Vertex Operator Algebra
  9. $\mathcal{N}=1$ Superconformal Vertex Operator Algebra And Spin Lifts
  10. Spin Lifts Of A Conformal Field Theory
  11. Formal Description Of Gauging A Discrete Symmetry
  12. The Arf Theory
  13. Coupling A CFT To The Arf Theory
  14. Superconformal Symmetry And The Moonshine Module
  15. The Beauty And The Beast Theory As The Spin Lift Of The FLM VOA
  16. ...and 12 more sections

Key Result

Proposition 2.1

Let $\left(\mathscr{H}, \mathscr{F}, \mathbf{V},|0\rangle, \psi_{L}\right)$ be an Hermitian vertex operator algebra. Then the following statements hold:

Theorems & Definitions (95)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.1
  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.2
  • Definition 2.6
  • ...and 85 more